Applied Mathematics Letters

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1 Applied Mathematics Letters 23 (2010) Contents lists available at ScienceDirect Applied Mathematics Letters journal homepage: The Roman domatic number of a graph S.M. Sheikholeslami a,b,, L. Volkmann c a Department of Mathematics, Azarbaijan University of Tarbiat Moallem, Tabriz, Islamic Republic of Iran b School of Mathematics, Institute for Research in Fundamental Sciences (IPM), P.O. Box: , Tehran, Islamic Republic of Iran c Lehrstuhl II für Mathematik, RWTH Aachen University, Aachen, Germany a r t i c l e i n f o a b s t r a c t Article history: Received 19 January 2010 Received in revised form 6 June 2010 Accepted 9 June 2010 Keywords: Roman dominating function Roman domination number Roman domatic number A Roman dominating function on a graph G is a labeling f : V(G) {0, 1, 2} such that every vertex with label 0 has a neighbor with label 2. A set { f 1, f 2,..., f d } of Roman d dominating functions on G with the property that f i(v) 2 for each v V(G) is called a Roman dominating family (of functions) on G. The maximum number of functions in a Roman dominating family on G is the Roman domatic number of G, denoted by d R (G). In this work we initiate the study of the Roman domatic number in graphs and we present some sharp bounds for d R (G). In addition, we determine the Roman domatic number of some graphs Elsevier Ltd. All rights reserved. 1. Introduction In this work, G is a simple graph with vertex set V = V(G) and edge set E = E(G). The order V of G is denoted by n = n(g). For every vertex v V, the open neighborhood N(v) is the set {u V(G) uv E(G)} and the closed neighborhood of v is the set N[v] = N(v) {v}. The degree of a vertex v V is d(v) = N(v). The minimum and maximum degree of a graph G are denoted by δ = δ(g) and = (G), respectively. If every vertex of G has degree k, then G is said to be k-regular. The open neighborhood of a set S V is the set N(S) = v S N(v), and the closed neighborhood of S is the set N[S] = N(S) S. The complement of a graph G is denoted by G. A tree is a connected acyclic graph and a cactus is a connected graph in which every block is an edge or a cycle. We write K n for the complete graph of order n and C n for a cycle of length n. For notation and graph theory terminology in general we follow [1] and [2]. A subset S of vertices of G is a dominating set if N[S] = V. The domination number γ (G) is the minimum cardinality of a dominating set of G. A domatic partition is a partition of V into dominating sets, and the domatic number d(g) is the largest number of sets in a domatic partition. The domatic number was introduced by Cockayne and Hedetniemi [3]. In their paper, they showed that γ (G) d(g) n. A Roman dominating function (RDF) on a graph G = (V, E) is defined in [4,5] as a function f : V {0, 1, 2} satisfying the condition that every vertex v for which f (v) = 0 is adjacent to at least one vertex u for which f (u) = 2. The weight of an RDF f is the value ω( f ) = v V f (v). The Roman domination number of a graph G, denoted by γ R(G), equals the minimum weight of an RDF on G. A γ R (G)-function is a Roman dominating function of G with weight γ R (G). A Roman dominating function f : V {0, 1, 2} can be represented by the ordered partition (V 0, V 1, V 2 ) (or (V f, 0 V f, 1 V f 2 ) to refer to f ) of V, (1) Corresponding author at: Department of Mathematics, Azarbaijan University of Tarbiat Moallem, Tabriz, Islamic Republic of Iran. Tel.: ; fax: addresses: s.m.sheikholeslami@azaruniv.edu (S.M. Sheikholeslami), volkm@math2.rwth-aachen.de (L. Volkmann) /$ see front matter 2010 Elsevier Ltd. All rights reserved. doi: /j.aml

2 1296 S.M. Sheikholeslami, L. Volkmann / Applied Mathematics Letters 23 (2010) where V i = {v V f (v) = i}. In this representation, its weight is ω(f ) = V V 2. Since V f 1 V f 2 is a dominating set when f is an RDF, and since placing weight 2 at the vertices of a dominating set yields an RDF, [6] observed that γ (G) γ R (G) 2γ (G). A set {f 1, f 2,..., f d } of distinct Roman dominating functions on G with the property that d f i(v) 2 for each v V(G) is called a Roman dominating family (of functions) on G. The maximum number of functions in a Roman dominating family (RD family) on G is the Roman domatic number of G, denoted by d R (G). The Roman domatic number is well-defined and d R (G) 1 (3) for all graphs G since the set consisting of any RDF forms an RD family on G. Our purpose in this work is to initiate the study of the Roman domatic number for graphs. We first study basic properties and bounds for the Roman domatic number of a graph. In addition, we determine the Roman domatic number of some classes of graphs. We start with the following observations and properties. Observation 1. If K n is the complete graph of order n 1, then d R (K n ) = n. Observation 2. If G is a graph, then d R (G) = 1 if and only if G is empty. Proof. If G is empty, then the mapping f : V(G) {0, 1, 2} defined by f (v) = 1 for each v V is the unique Roman dominating function on G and so d R (G) = 1. Conversely, let E(G) and let uv E(G). Then the mappings and f : V(G) {0, 1, 2} defined by f (u) = 2, f (v) = 0 and f (x) = 1 for each x V {u, v} g : V(G) {0, 1, 2} defined by g(u) = 0, g(v) = 2 and g(x) = 1 for each x V {u, v} are Roman dominating functions on G and {f, g} is a Roman dominating family on G. It follows that d R (G) 2, and the proof is complete. Proposition A ([6]). For k 1, 1. γ R (C 3k ) = 2k, 2. γ R (C 3k+1 ) = 2k + 1, 3. γ R (C 3k+2 ) = 2k + 2. Proposition B ([7]). If G is a graph of order n, then γ R (G) n (G) + 1. (2) 2. Properties of the Roman domatic number In this section we present basic properties of d R (G) and sharp bounds on the Roman domatic number of a graph. Theorem 3. Let G be a graph of order n with Roman domination number γ R (G) and Roman domatic number d R (G). Then γ R (G) d R (G) 2n. Moreover, if γ R (G) d R (G) = 2n, then for each RD family {f 1, f 2,..., f d } on G with d = d R (G), each function f i is a γ R (G)-function and d f i(v) = 2 for all v V. Proof. Let {f 1, f 2,..., f d } be an RD family on G such that d = d R (G) and let v V. Then d γ R (G) = γ R (G) f i (v) v V = v V f i (v) 2 v V = 2n.

3 S.M. Sheikholeslami, L. Volkmann / Applied Mathematics Letters 23 (2010) If γ R (G) d R (G) = 2n, then the two inequalities occurring in the proof become equalities. Hence for the RD family {f 1, f 2,..., f d } on G and for each i, v V f i(v) = γ R (G); thus each function f i is a γ d R (G)-function, and f i(v) = 2 for all v V. Let A 1 A 2... A d be a domatic partition of V(G) into dominating sets such that d = d(g). Then the set of functions {f 1, f 2,..., f d } with f i (v) = 2 if v A i and f i (v) = 0 otherwise for 1 i d is an RD family on G. This shows that d(g) d R (G) for every graph G. Since γ R (G) 2 for each graph G of order n 2, Theorem 3 implies that d R (G) n. Combining these two observations, we obtain the following result. Corollary 4. For any graph G of order n, d(g) d R (G) n. Proposition 5. Let G be a graph of order n 2. Then γ R (G) = n and d R (G) = 2 if and only if (G) = 1. Proof. Let γ R (G) = n and d R (G) = 2. It follows from Proposition B that (G) 1. Since d R (G) = 2, we have E(G) and so (G) 1. Thus (G) = 1. Conversely, let (G) = 1. Then G = rk 1 n r 2 K 2 with r n 2, and we have γ R (G) = rγ R (K 1 ) + n r 2 γ R(K 2 ) = r + (n r) = n. By Theorem 3 and Observation 2, we obtain d R (G) = 2. This completes the proof. Proposition 6. If G is a graph of order n 2, then d R (G) = n if and only if G is the complete graph on n vertices. Proof. If G is the complete graph on n vertices, then the result follows by Observation 1. Conversely, let d R (G) = n. If n = 2, then the result is immediate. Assume n 3. Then γ R (G) 2 and it follows by Theorem 3 that γ R (G) = 2. Let { f 1, f 2,..., f n } be an RD family on G and let V(G) = {v 1, v 2,..., v n }. Then by Theorem 3, f i is a γ d R (G)-function for each i, and f i(v) = 2 for all v V(G). Since n 3 and γ R (G) = 2, we conclude that for each i, there exists an index 1 i j n such that f i (v ij ) = 2 and f i (v k ) = 0 for v k V(G) {v ij }. On the other hand, since d f i(v) = 2 for all v, i j k j if i k. It follows that {1 j, 2 j,..., n j } = {1, 2,..., n} which implies that each v i is adjacent to all other vertices. Thus G is the complete graph on n vertices. Theorem 7. If G is a graph of order n 2, then γ R (G) + d R (G) n + 2 (4) with equality if and only if (G) = 1 or G is a complete graph. Proof. If d R (G) 1, then obviously γ R (G) + d R (G) n + 1. Let now d R (G) 2. Since γ R (G) 2, we have d R (G) n. According to Theorem 3, we obtain γ R (G) + d R (G) 2n d R (G) + d R(G). Using the fact that the function g(x) = x + (2n)/x is decreasing for 2 x 2n and increasing for 2n x n, this inequality leads to the desired bound immediately. If G is the complete graph on n vertices, then obviously γ R (G) = 2 and by Observation 1, d R (G) = n. If (G) = 1, then by Proposition 5, we have γ R (G) = n and d R (G) = 2. Thus in all cases γ R (G) + d R (G) = n + 2. Conversely, let equality hold in (4). It follows from (5) that n + 2 = γ R (G) + d R (G) 2n d R (G) + d R(G) n + 2, which implies that γ R (G) = 2n d R (G) and d R(G) = 2 or d R (G) = n. If d R (G) = n, then G is a complete graph by Proposition 6. If d R (G) = 2, then γ R (G) = n, and it follows from Proposition 5 that (G) = 1. This completes the proof. Using Proposition A and Theorem 3, we determine the Roman domatic number of cycles. Proposition 8. If C n is the cycle on n 3 vertices, then { 2 if n 1, 2 (mod 3) d R (C n ) = 3 if n 0 (mod 3). Proof. Let {v 1, v 2,..., v n } be the vertex set of C n. First let n 0 (mod 3). Then n = 3k for some k 1. By Proposition A and Theorem 3, we have d R (C n ) 3. Define the Roman dominating functions f 1, f 2, f 3 as follows: f j (v 3(i 1)+j ) = 2 for 0 i n/3 1, 1 j 3 and f j (x) = 0 otherwise, where the indices are taken modulo 3. It is easy to see that f i is a Roman dominating function on G for each i and {f 1, f 2, f 3 } is a Roman dominating family on G. Thus d R (C n ) = 3. (5)

4 1298 S.M. Sheikholeslami, L. Volkmann / Applied Mathematics Letters 23 (2010) Now let n 1 (mod 3). Then n = 3k + 1 for some k 1. By Proposition A and Theorem 3, we have d R (C n ) 2. Applying Observation 2, it follows that d R (C n ) = 2. Finally, let n 2 (mod 3). Then n = 3k + 2 for some k 1, and as above, we obtain the desired result d R (C n ) = 2. Theorem 9. For every graph G, d R (G) δ(g) + 2. Moreover, the upper bound is sharp. Proof. If d R (G) 2, the result is immediate. Let now d R (G) 3 and let {f 1, f 2,..., f d } be an RD family on G such that d = d R (G). Assume that v is a vertex of minimum degree δ(g). Since the equality u N[v] f i(u) = 1 holds for at most two indices i {1, 2,..., d}, we have 2d 2 u N[v] = u N[v] u N[v] 2 f i (u) f i (u) = 2(δ(G) + 1). Thus d R (G) δ(g) + 2. To prove sharpness, let G i be a copy of K k+3 with vertex set V(G i ) = {v i, 1 vi,..., 2 vi k+3 } for 1 i k and let the graph G be obtained from k G i by adding a new vertex v and joining v to each v i 1. Define the Roman dominating functions f 1, f 2,..., f k+2 as follows: and f i (v i ) = 2, 1 f i(v j i+1 ) = 2 if j {1, 2,..., k} {i} and f (x) = 0 otherwise (1 i k), f k+1 (v) = 1, f k+1 (v j k+2 ) = 2, if j {1, 2,..., k} and f (x) = 0 otherwise, f k+2 (v) = 1, f k+2 (v j k+3 ) = 2, if j {1, 2,..., k} and f (x) = 0 otherwise. It is easy to see that f i is a Roman dominating function on G for each i and {f 1, f 2,..., f k+2 } is a Roman dominating family on G. Since δ(g) = k, we have d R (G) = δ(g) + 2. For regular graphs we will give a better upper bound on d R (G). For the proof we make use of the following result, which generalizes a known lower bound on γ R (G), given in the article [8]. Theorem 10. Let G be a graph of order n and maximum degree 1. Then 2n γ R (G) + ɛ + 1 with ɛ = 0 when n 0, 1(mod ( + 1)) and ɛ = 1 when n 0, 1, (mod ( + 1)). Proof. Let n = p( + 1) + r with integers p 1 and 0 r, and let f = (V 0, V 1, V 2 ) be a γ R (G)-function. Then γ R (G) = V V 2 and n = V 0 + V 1 + V 2. Since each vertex of V 0 is adjacent to at least one vertex of V 2, we deduce that V 0 V 2. Therefore we conclude that ( + 1)γ R (G) = ( + 1)( V V 2 ) = ( + 1) V V V 2 ( + 1) V V V 0 = 2n + ( 1) V 1 = 2p( + 1) + 2r + ( 1) V 1. This inequality chain and the hypothesis that 1 lead to the desired bound if r = 0 or r = 1 or 2 r and V 1. In the remaining case where 2 r and V 1 =, it follows from V 0 V 2 that p( + 1) + r = n = V 0 + V 2 ( + 1) V 2. Hence the condition r 2 leads to V 2 p + 1. Therefore we obtain γ R (G) = 2 V 2 2(p + 1), and this completes the proof.

5 S.M. Sheikholeslami, L. Volkmann / Applied Mathematics Letters 23 (2010) Corollary 11 ([8]). If G is a graph of order n and maximum degree 1, then 2n γ R (G). + 1 Theorem 12. If G is a δ-regular graph of order n, then d R (G) δ + ɛ with ɛ = 1 when n 0(mod (δ + 1)) and ɛ = 0 when n 0(mod (δ + 1)). Proof. If δ = 0, then Observation 2 implies the desired result. Let now δ 1, n = p(δ + 1) + r with integers p 1 and 0 r δ, and let {f 1, f 2,..., f d } be an RD family on G such that d = d R (G). It follows that ω(f i ) = f i (v) = v V v V f i (v) 2 = 2n. (7) v V If r = 0, then we deduce from Theorem 10 that ω(f i ) γ R (G) 2p for each i {1, 2,..., d}. Suppose to the contrary that d δ + 2. Then we obtain ω(f i ) 2pd 2p(δ + 2) > 2n. This is a contradiction to (7) and thus d δ + 1. If r = 1, then Theorem 10 implies that ω(f i ) γ R (G) 2p + 1 for each i {1, 2,..., d} and δ 2. Suppose to the contrary that d δ + 1. Then we obtain the contradiction ω(f i ) d(2p + 1) (δ + 1)(2p + 1) > 2n. Therefore d δ, and (6) is proved in this case. Finally assume that 2 r δ. Then Theorem 10 yields ω( f i ) γ R (G) 2p + 2 for each i {1, 2,..., d}. If we suppose to the contrary that d δ + 1, then we obtain ω(f i ) d(2p + 2) (δ + 1)(2p + 2) > 2n. This contradiction to (7) implies that d δ in this case, and the proof of Theorem 12 is complete. Using Theorem 12 instead of Proposition A and Theorem 3, one can prove Proposition 8 by a method analogous to that above. As an application of Theorems 9 and 12, we will prove the following Nordhaus Gaddum type results. Theorem 13. For every graph G of order n, (6) d R (G) + d R (G) n + 2. (8) Proof. It follows from Theorem 9 that d R (G) + d R (G) (δ(g) + 2) + (δ(g) + 2) = (δ(g) + 2) + (n (G) 1 + 2). If G is not regular, then (G) δ(g) 1, and hence this inequality implies the desired bound d R (G) + d R (G) n + 2. If G is δ(g)-regular, then we deduce from Theorem 12 that d R (G) + d R (G) (δ(g) + 1) + (δ(g) + 1) = (δ(g) + 1) + (n δ(g) 1 + 1) = n + 1, and the proof of the Nordhaus Gaddum bound (8) is complete. Theorem 12 and the proof of Theorem 13 demonstrate that for regular graphs the following better Nordhaus Gaddum inequality is valid. Theorem 14. If G is a δ(g)-regular graph of order n, then d R (G) + d R (G) n + 1, and equality in (9) implies n 0 (mod (δ(g) + 1)) and n 0 (mod (δ(g) + 1)). (9)

6 1300 S.M. Sheikholeslami, L. Volkmann / Applied Mathematics Letters 23 (2010) If G is isomorphic to the complete graph K n, then d R (G) = n and d R (G) = 1 and therefore d R (G) + d R (G) = n + 1. This example demonstrates that the bound (9) is sharp. Theorem 15. For any tree T of order n 2, d R (T) = 2. Proof. By Observation 2, it suffices to show that d R (T) 2. If diam(t) 2, the result is immediate. Let diam(t) 3 and let v 1 v 2... v diam(t) be a diametral path in T. Assume that v 1 = u 1, u 2,..., u r are the leaves adjacent to v 2, and assume that {f 1, f 2,..., f d } is an RD family on T such that d = d R (T). Claim. If f i (u j ) = 2 for some i and j, then d 2. Proof of Claim. Let, without loss of generality, f 1 (v 1 ) = 2. Then f i (v 1 ) = 0 for each 2 i d. Since f i is a Roman dominating function we must have f i (v 2 ) = 2 for 2 i d d. Now the claim follows from f i(v 2 ) 2. Thus we assume that f i (u j ) 1 for each i and j. Since f i is a Roman dominating function and d 2, we have (f i (v 2 ) = 2 or f i (v 3 ) = 2 for each i) or (f i (v 2 ) = f i (v 3 ) = 1 for some i, if d = 2). If d d 3, then obviously f i(v 2 ) d 3 or f i(v 3 ) 3 which is a contradiction. Thus d 2 and the proof is complete. Theorem 16. If G is a cactus graph, then d R (G) 3. Proof. Let d = d R (G). If δ(g) 1, then Theorem 9 implies the desired bound d R (G) 3. It remains the case that δ(g) = 2. If G consists of a collection of vertex disjoint cycles, then the result follows from Proposition 8. Otherwise, the cactus graph G contains a cycle C t = v 1 v 2... v t v 1 with exactly one cut vertex, say v 1, of G. Claim. If f i (v j ) = 2 for some i and 2 j t, then d 3. Proof of Claim. Let, without loss of generality, f 1 (v 2 ) = 2. Then f i (v 2 ) = 0 for each 2 i d. Since f i is a Roman dominating function we must have f i (v 1 ) = 2 or f i (v 3 ) = 2 for 2 i d d. Now the claim follows from f i(x) 2 for each vertex x V(G). Thus we assume that f i (v j ) 1 for each i and 2 j d. If f i (v 1 ) 1 for each i, then f i (v 2 ) = 1 for each i, and we deduce that d 2. Finally assume, without loss of generality, that f 1 (v 1 ) = 2. This implies that f i (v 1 ) = 0 for each 2 i d and thus f i (v 2 ) = 1 for each 2 i d d. Since f i(v 2 ) 2, it follows that d 3, and the proof is complete. The cycles of length 3p show that Theorem 16 is sharp. However there exist a lot of further cactus graphs G with d R (G) = 3. Acknowledgement This research was in part supported by a grant from IPM (No ). References [1] T.W. Haynes, S.T. Hedetniemi, P.J. Slater, Fundamentals of Domination in Graphs, Marcel Dekker, Inc., New York, [2] D.B. West, Introduction to Graph Theory, Prentice-Hall, Inc., [3] E.J. Cockayne, S.T. Hedetniemi, Towards a theory of domination in graphs, Networks 7 (1977) [4] C.S. Revelle, K.E. Rosing, Defendens imperium romanum: a classical problem in military strategy, Amer. Math. Monthly 107 (7) (2000) [5] I. Stewart, Defend the Roman Empire, Sci. Amer. 281 (6) (1999) [6] E.J. Cockayne, P.M. Dreyer Jr., S.M. Hedetniemi, S.T. Hedetniemi, On Roman domination in graphs, Discrete Math. 278 (2004) [7] E.W. Chambers, B. Kinnersley, N. Prince, D.B. West, Extremal problems for Roman domination, SIAM J. Discr. Math. 23 (3) (2009) [8] E.J. Cockayne, P.J.P. Grobler, W.R. Gründlingh, J. Munganga, J.H. van Vuuren, Protection of a graph, Util. Math. 67 (2005)